LECTURE 3: Polynomial interpolation and numerical differentiation

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Example The Richardson extrapolation algorithm is implemented in the following procedure Derivative: void Derivative(double (*f)(double x), double x, int n, double h, double D[][NCOLS]) { int i, j; double hh; } hh=h; for(i=0; i
4.5 First-derivative formulas via interpolation polynomials An important stratagem can be used to approximate derivatives (as well as integrals and other quantities). The function f is first approximated by a polynomial p. Then we simply approximate the derivative of f by the derivative of p: f ′ (x)≈ p ′ (x). Of course, we must be careful with this strategy because the behavior of p may be oscillatory. In practice, the approximating polynomial is often determined by interpolation at a few points. For two nodes, we have Consequently, p 1 (x)= f(x 0 )+ f[x 0 ,x 1 ](x−x 0 ) f ′ (x)≈ p ′ (x)= f[x 0 ,x 1 ]= f(x 1)− f(x 0 ) x 1 − x 0 If x 0 = x and x 1 = x+h, this formula gives If x 0 = x−h and x 1 = x+h, we get f ′ (x)≈ 1 [ f(x+h)− f(x)] h f ′ (x)≈ 1 [ f(x+h)− f(x−h)] 2h Now consider interpolation at three nodes The derivative is p 2 (x)= f(x 0 )+ f[x 0 ,x 1 ](x−x 0 )+ f[x 0 ,x 1 ,x 2 ](x−x 0 )(x−x 1 ) p ′ 2(x)= f[x 0 ,x 1 ]+ f[x 0 ,x 1 ,x 2 ](2x−x 0 − x 1 ) Here the first term is the same as in the previous case and the second term is a refinement or a correction term. The accuracy of the leading term depends on the choice of the nodes x 0 and x 1 . If we have x=(x 0 + x 1 )/2, then the correction term is zero. Thus the first term must be more accurate than in other cases. This is why choosing a central difference x 0 = x−h and x 1 = x+h gives more accurate results than x 0 = x and x 1 = x+h.
Page 1 and 2: LECTURE 3: Polynomial interpolation
Page 3 and 4: Note that when l i (x) is evaluated
Page 5 and 6: 2.4 Newton form The inductive proof
Page 7 and 8: 2.5 Nested multiplication For effic
Page 9 and 10: Example. Determine the quantities f
Page 11 and 12: 2.7 Recursive property of divided d
Page 13 and 14: The table with a completed second c
Page 15 and 16: The following procedure Coefficient
Page 17 and 18: Figure 1 shows a plot of the data p
Page 19 and 20: 2.9 Inverse interpolation A process
Page 21 and 22: We can simplify the notation by S i
Page 23 and 24: 3.1 Choice of nodes Contrary to int
Page 25 and 26: Example. In a previous example, we
Page 27 and 28: 4 Numerical differentiation We now
Page 29: 4.3 Richardson extrapolation theore
Page 33 and 34: This allows us to write out the for

LECTURE 3: Polynomial interpolation and numerical differentiation

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